"Stephen last week had a potentially cancerous tumour removed from his jaw line in front of his left ear, and yesterday received the very good news that the tumour was benign," the New Zealand team general manager Lindsay Crocker said Friday.
"The size of the tumour caused nerve damage, which was repaired with further surgery on Wednesday."
Spin bowler Daniel Vettori will captain New Zealand against Australia in Auckland on December 3 and in Wellington on December 7, and Fleming is likely to return for the final match in Christchurch on December 10, New Zealand Cricket said.
Batsman Nathan Astle will be vice-captain for New Zealand, which named an expanded 14-man squad on Wednesday in case Fleming would not be available.
The New Zealand Herald quoted Vettori saying he was looking forward to measuring his captaincy abilities against the world's top one-day side.
"I'm getting my head around the whole situation," he said.
"We've had Fleming in the line-up for so long now and the thought of not having him for one of our biggest series is a big change, but one that I'm sure we'll all adapt to," he said.
Fleming has been New Zealand's most successful captain since taking over in 1997.